51 Implication in detail (Atheism - Constructing a Logical Argument)
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This article is from the Atheism FAQ, by mathew meta@pobox.com with numerous contributions by others.
51 Implication in detail (Atheism - Constructing a Logical Argument)
There's one very important thing to remember:
The fact that a deductive argument is valid doesn't necessarily
mean that its conclusion holds.
That may seem confusing, but it's because of the slightly
counter-intuitive nature of how implication works.
Obviously you can build a valid argument out of true propositions. But
you can also build a completely valid argument using only false
propositions. For example:
* All insects have wings (premise)
* Woodlice are insects (premise)
* Therefore woodlice have wings (conclusion)
The conclusion isn't true because the argument's premises are false.
If the argument's premises were true, however, the conclusion would be
true. So the argument is entirely valid.
More subtly, you can reach a true conclusion from false premises --
even ludicrously false ones:
* All fish live in the ocean (premise)
* Sea otters are fish (premise)
* Therefore sea otters live in the ocean (conclusion)
However, there's one thing you can't do: start with true premises, go
through a valid deductive argument, and arrive at a false conclusion.
(Remember the definition of a valid deductive argument.)
So, here's a "truth table" for implication. The symbol "=> " denotes
implication; "A" is the premise, "B" the conclusion. "T" and "F"
represent true and false respectively.
CAPTION: Truth Table for Implication
Premise Conclusion Inference
A B A => B
false false true
false true true
true false false
true true true
* If the premises are false and the inference valid, the conclusion
can be true or false. (Lines 1 and 2.)
* If the premises are true and the conclusion false, the inference
must be invalid. (Line 3.)
* If the premises are true and the inference valid, the conclusion
must be true. (Line 4.)
A sound argument is a valid argument whose premises are true. A sound
argument therefore arrives at a true conclusion. Be careful not to
confuse sound arguments with valid arguments.
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